Time triggered stochastic hybrid system with nonlinear continuous dynamics

Abstract

Time triggered stochastic hybrid systems (TTSHS) constitute a class of piecewise-deterministic Markov processes (PDMP), where continuous-time evolution of the state space is interspersed with discrete stochastic events. Whenever a stochastic event occurs, the state space is reset based on a random map. Prior work on this topic has focused on the continuous-time evolution being modeled as a linear time- invariant system, and in this contribution, we generalize these results to consider nonlinear continuous dynamics. Our approach relies on approximating the nonlinear dynamics between two successive events as a linear time-varying system and using this approximation to derive analytical solutions for the state space’s statistical moments. The TTSHS framework is used to model continuous growth in an individual cell’s size and its subsequent division into daughters. It is well known that exponential growth in cell size, together with a size- independent division rate, leads to an unbounded variance in cell size. Motivated by recent experimental findings, we consider nonlinear growth in cell size based on a Michaelis- Menten function and show that this leads to size homeostasis in the sense that the variance in cell size remains bounded. Moreover, we provide a closed-form expression for the variance in cell size as a function of model parameters and validate it by performing exact Monte Carlo simulations. In summary, our work provides an analytical approach for characterizing moments of a nonlinear stochastic dynamical system that can have broad applicability in studying random phenomena in both engineering and biology.

Using higher-order statistics to infer random forces driving a nano sensor

“…Another interesting avenue is to investigate feedback strategies, such as applying an external force dependent on displacement and/or velocity for mitigating noise in nanosensors. We have recently extended the theory of time-triggered SHS to nonlinear dynamics systems [61], which can be used to explore the effects of nonlinearities in beam deformation and physical collisions on state fluctuations.…”

Stochastic analysis of frequency bandwidth and noise attenuation in neurotransmission

“…3, where CV 2 b increases with f . In the limit f → ∞, we substitute F from ( 24) in (28), where where v m = lim f →∞ v max . Now further assuming v th v m , and using the fact that x 2 /(1 − x) 2 / ln 2 (1 − x) ≈ 1 + x for small x, (29) reduces to…”

“…Here we apply the formulas of Stochastic Hybrid Systems (SHS) that effectively combine discrete and continuous random processes [14]- [28] to investigate how stochasticity in neurotransmitter release impacts the timing of postsynaptic AP generation via the LIF model. This work builds on our previous work that modeled the presynaptic vesicle turnover [29]- [30] to understand the downstream impact of vesicles dynamics on postsynaptic processes.…”

“…Fig.6.The postsynaptic AP frequency F and noise in AP timing CV2 T for frequency-dependent parameters. Plots of frequency F as given in(23) and noise(28) for fixed probability of release and vesicle refilling rates are compared with the case of frequency-dependent parameters as phenomenologically captured via(32). Parameters are chosen as k = 5 sec −1 , pmax = 0.54, F 1 = 10 Hz, h 1 = 1.41, kmax = 20 sec −1 , F 2 = 10 Hz, h 2 = 1.56 and pr = 0.3, M = 100, v 0 = 0 volts, v th = 0.07 volts, τv = 10 sec, kv = 0.001 volts.…”